Question: If $x$, $y$, and $z$ are positive with $xy=20\sqrt[3]{2}$, $xz = 35\sqrt[3]{2}$, and $yz=14\sqrt[3]{2}$, then what is $xyz$?
Answer: Taking the product of the equations gives  \begin{align*}
xy\cdot xz\cdot yz &= 20\sqrt[3]{2} \cdot 35\sqrt[3]{2} \cdot 14\sqrt[3]{2}\\
(xyz)^2 &= 2^4\cdot5^2\cdot7^2\\
xyz &= 2^2\cdot5\cdot7 = \pm 140
\end{align*} Since we were given that $x$, $y$, and $z$ are positive, we can conclude that $xyz = \boxed{140}$.